Part 2:
So my advice:
0) Figure out if your baby is ready for sleep training. Only you would know that. In my case, with both my babies, we waited until they were 7-8 months old.
1) You and your partner sit down and agree on a sleep training method AND STICK TO IT NO MATTER WHAT. FOr me, the best is extinction, where you leave them in bed, turn around and leave and not enter the room anymore, no matter how much they cry. Some people prefer the ferber method thinking it's gentler, where they check in with the baby at scheduled intervals (2, 4, 6/6/6/... the first night, 3/5/10/10/10... the 2nd, etc.) but without touching them and just to say from afar 'it's ok, we're here, go to sleep'. But to me, that just aggravates the baby more.
2) Figure out who's doing what. If the dad is stronger in this, then unfortunately he has to do it, and the mum can take a walk for an hour every night for a week.
3) You make sure the nap schedule and wake windows are correct. Too little nap, too much nap, the last nap before sleep being too far away or too close to bedtime, and it'll screw everything up. And you'll end up making the baby cry for nothing. AND YOU STICK WITH THE SCHEDULE. For at least 2 weeks, those dinner plans with friends where the baby sleeps in the stroller, or anything else, they get put on hold. #1 priority is sticking to the baby's schedule.
4) Make sure the baby's getting plenty of activity during the wake windows. A lot of people also prefer to have quiet time an hour before bedtime. I actually think it's better to throw the baby up and down and have some fun "wrestling" and get physical with the baby 30-45 minutes before bedtime, then start the calm time.
5) You devise a bedtime routine (dinner, shower with calming music, lots of kisses and cuddles in bed, milk/formula, bedtime stories, some gentle rocking with dimmed lights, then leaving the baby in bed, walking out of the bedroom, and NOT COMING BACK IN) and... you guess it, you STICK WITH IT. Same routine, same bedtime stories, night after night.
6) If the baby wakes up in the middle of the night, you give it AT LEAST 5 minutes before you go in, if for example you're doing the Ferber method. You have to be consistent.
Speaking of being consistent, some general points:
* Consistency: if you're not consistent, it's not going to work. The baby's just going to get confused if one night they cry themselves to sleep, another night you help them to sleep, etc.
* Not sticking to the plan: worst thing you can do. For example, you let the baby cry, it's been 20 minutes, you can't take it anymore, and then you go in and rock them to sleep. That way, they'll just learn they have to cry until you come in.
* I advice that they be done with any food/milk at least 30 minutes before bedtime, especially if you have REALLY LOUD screamers like mine that get very anxious and start pulling their hair and inhale a lot of air in the process of hysterically screaming and crying. Because they can end up taking in a lot of air and then they puke (both of mine did on the first or second night of training).
* Your sleep situation is not unique, and you're no worse than most parents. Trust me. We all think our kids our the worst when it comes to sleeping, but that's just not the case.
If you want to wait for her to be a bit older, what you can do is stick to a plan like putting her down when she's drowsy and awake, but then tapping her back or bum and trying to put her back down (if she sits/stands up) until she falls asleep. She *WILL* cry a lot, but you're there with her and so maybe that's easier for you. Then when they're 7-8 months older and the schedules are all set and the routines are fixed and they're taking in a good amount of solids, you can try a more strict approach.
And if you let them cry, just remember: her belly is full, her diaper is clean, she's healthy, etc. So this crying is just because she can't express herself in any other way to say 'wait a minute, this is not how i've been going to sleep for 5 months'.
Good luck.